Short-depth active learning quantum amplitude estimation without eigenstate collapse

ABSTRACT

Techniques and a system to facilitate estimation of a quantum phase, and more specifically, to facilitate estimation of an expectation value of a quantum state, by utilizing a hybrid of quantum and classical methods are provided. In one example, a system is provided. The system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can include an encoding component and a learning component. The encoding component can encode an expectation value associated with a quantum state. The learning component can utilize stochastic inference to determine the expectation value based on an uncollapsed eigenvalue pair.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under Contract No.:FA8750-18-C-0098, awarded by U.S. Air Force, Office of ScientificResearch (AFOSR). The Government has certain rights to this invention.

BACKGROUND

The subject disclosure relates to estimation of a quantum state's phase,and more specifically, to estimation of a quantum state's phaseutilizing the estimation of the amplitude between two quantum states,and more specifically, to the estimation of the amplitude utilizing ahybrid of quantum and classical methods.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, devices, systems, computer-implemented methods,apparatus and/or computer program products facilitating estimation of aquantum state's phase, and more specifically, facilitating estimation ofa quantum state phase utilizing a hybrid of quantum and classicalmethods.

According to an embodiment, a system is provided. The system cancomprise a memory that stores computer executable components and aprocessor that executes the computer executable components stored in thememory. The computer executable components can include learningcomponent that can utilize stochastic inference to determine anexpectation value based on an uncollapsed eigenvalue pair. The computerexecutable components can further include an encoding component that canencode the expectation value associated with a quantum state.

According to an embodiment, a system is provided. The system cancomprise a memory that stores computer executable components and aprocessor that executes the computer executable components stored in thememory. The computer executable components can include a learningcomponent. The learning component can utilize stochastic inference todetermine an expectation value based on an uncollapsed eigenvalue pair.The learning component can include a measuring component that canprobabilistically measure the expectation value, with the measuringcomponent being independent of an input state.

According to another embodiment, a computer-implemented method isprovided. The computer-implemented method can comprise learning, by asystem operatively coupled to a processor, an expectation value based ona quantum state based on an uncollapsed eigenvalue pair, by stochasticinference. In some embodiments, the learning can comprise measuring, bythe system, the expectation value probabilistically, with the measuringbeing independent of an input state. The method can further compriseencoding, by the system, an expectation value associated with a quantumstate, as a phase.

According to another embodiment, a computer-implemented method isprovided. The computer-implemented method can comprise encoding, by asystem operatively coupled to a processor an expectation valueassociated with a quantum state. The method can further compriselearning, by the system, by stochastic inference, the expectation valuebased on an uncollapsed eigenvalue pair.

According to yet another embodiment, a computer program productfacilitating phase estimation of a quantum state can comprise a computerreadable storage medium having program instructions embodied therewith.The program instructions can be executable by a processor and cause theprocessor to encode, by the processor, an expectation value associatedwith a quantum state. Further, the program instructions can cause theprocessor to learn, by utilizing a stochastic inference, the expectationvalue based on an uncollapsed eigenvalue pair.

DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of an example, non-limiting systemthat can generate an eigenvalue from a quantum state in accordance withone or more embodiments described herein.

FIG. 2A illustrates an example approach that can use a circuit to selecta control parameter m that can be used by an encoding component toencode an expectation value in the phase of a quantum state, inaccordance with one or more embodiments.

FIG. 2B illustrates a block diagram of an example, non-limiting systemthat shows the interaction of the encoding component and learningcomponent in accordance with one or more embodiments described herein.

FIG. 3 illustrates an example, non-limiting circuit that shows anencoding component in accordance with one or more embodiments describedherein.

FIG. 4 illustrates an example, non-limiting geometric representation ofthe example, non-limiting circuit of FIG. 3 in accordance with one ormore embodiments described herein.

FIG. 5 illustrates a block diagram of an example, non-limiting systemthat can generate an eigenvalue from a quantum state in accordance withone or more embodiments described herein.

FIG. 6 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can generate an eigenvalue from aquantum state in accordance with one or more embodiments describedherein.

FIG. 7 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can encode a quantum state as a phasein accordance with one or more embodiments described herein.

FIG. 8 illustrates a flow diagram of a computer-implemented method thatcan learn an eigenvalue corresponding to an expectation value based on aquantum state in accordance with one or more embodiments describedherein.

FIG. 9 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can generate an eigenvalue from aquantum state in accordance with one or more embodiments describedherein.

FIG. 10 illustrates a block diagram of an example, non-limitingoperating environment in which one or more embodiments described hereincan be facilitated.

FIG. 11 illustrates an example, non-limiting graph showing an exampleprobabilistic measurement of a phase in accordance with one or moreembodiments described herein.

FIGS. 12A and 12B illustrate example, non-limiting graphs showing thesamples for a given error E and circuit depth for a given number ofqubits.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details.

Quantum computing employs quantum physics principles such as thesuperposition principle and entanglement to encode information ratherthan binary digital techniques. For example, a quantum circuit canemploy quantum bits (e.g., qubits) that operate according to thesuperposition principle of quantum physics and the entanglementprinciple of quantum physics. The superposition principle of quantumphysics allows each qubit to represent both a value of “1” and a valueof “0” at the same time. The entanglement principle of quantum physicsstates allows qubits in a superposition to be correlated with eachother. For instance, a state of a first value (e.g., a value of “1” or avalue of “0”) can depend on a state of a second value. As such, aquantum circuit can employ qubits to encode information rather thanbinary digital techniques based on transistors. However, design of aquantum circuit can be generally difficult and/or time consuming ascompared to conventional binary digital devices. Furthermore, it isgenerally desirable to increase efficiency of a quantum circuit and/or aquantum computing process. As such, design of a quantum circuit and/orquantum computing processing can be improved.

To address these and/or other issues, embodiments described hereininclude systems, computer-implemented methods, and computer programproducts facilitating quantum computation. In one embodiment, providedis a system comprising a memory that stores computer executablecomponents, and a processor that executes the computer executablecomponents stored in the memory, wherein the computer executablecomponents comprise an encoding component that encodes an expectationvalue in the phase of a quantum state; and a learning component thatutilizes stochastic inference to determine an eigenvalue correspondingto the expectation value. The system can include a quantum processor. Inan aspect, the encoding component can encode the expectation value as aphase. According to certain embodiments, the encoding component canencode the expectation value based on an amplitude of the expectationvalue. In some embodiments, the stochastic inference can employ Bayesianlearning. The learning component can include a measuring component. Themeasuring component can utilize at least one ancilla qubit to determinethe eigenvalue corresponding to the expectation value. The measuringcomponent can produce an output comprising a probabilistic measurement.The probabilistic measurement can be based on the at least one ancillaqubit.

In another embodiment, provided is a system comprising a memory thatstores computer executable components, and a processor that executes thecomputer executable components stored in the memory, wherein thecomputer executable components comprise a learning component thatutilizes stochastic inference to determine an eigenvalue correspondingto an expectation value based on a the phase of a quantum state, whereinthe learning component comprises a measuring component thatprobabilistically measures the expectation value, wherein the measuringcomponent is independent of an input state. The system can include aquantum processor. The system can further comprise an encoding componentthat encodes the expectation value based on the phase of a quantumstate. The encoding component can encode the expectation value as aphase. In some embodiments, the encoding component can encode theexpectation value based on an amplitude of the expectation value. Insome embodiments, the stochastic inference can comprise Bayesianlearning, which can include Bayesian inference. According to certainembodiments, the measuring component can utilize at least one ancillaqubit to determine the eigenvalue corresponding to the expectationvalue.

In another embodiment, provided is a computer-implemented methodcomprising encoding, by a system operatively coupled to a processor, anexpectation value associated with the phase of a quantum state; andlearning, by the system, an eigenvalue corresponding to the expectationvalue by stochastic inference. The system can include a quantumprocessor. According to certain embodiments, the expectation value canbe encoded as a phase. In some embodiments, the expectation value can beencoded based on an amplitude of the expectation value. In an aspect,the stochastic inference can include Bayesian learning, which caninclude Bayesian inference. Learning an eigenvalue can includemeasuring, by the system, the eigenvalue corresponding to theexpectation value by utilizing at least one ancilla qubit. Measuring canproduce an output comprising a probabilistic measurement, wherein theprobabilistic measurement is based on the at least one ancilla qubit.

In another embodiment, provided is a computer-implemented methodcomprising: learning, by a system operatively coupled to a processor, aneigenvalue corresponding to an expectation value based on a quantumstate by stochastic inference, wherein learning comprises measuring, bythe system, the expectation value probabilistically, wherein themeasuring is independent of an input state. The system can include aquantum processor. The method can further include encoding, by thesystem, the expectation value as a phase. The encoding can be based onan amplitude of the expectation value. The stochastic inference caninclude Bayesian learning, which can include Bayesian inference.

FIG. 1 illustrates a block diagram of an example, non-limiting systemthat can generate an eigenvalue from a quantum state in accordance withone or more embodiments described herein. According to multipleembodiments, the system 150 can include memory 165 can store one or morecomputer and/or machine readable, writable, and/or executable componentsand/or instructions that, when coupled with a processor 160 and/orquantum processor 162, can facilitate performance of operations definedby the executable components and/or instructions. The quantum processor162 can be a machine that performs a set of calculations based onprinciples of quantum physics. For instance, the quantum processor 162can perform one or more quantum computations employing a set of quantumcircuits. Furthermore, the quantum processor 162 can encode informationusing qubits which can store information in superposition. Memory 165can store computer and/or machine readable, writable, and/or executablecomponents and/or instructions that, when executed by a processor 160 orquantum processor 162, can facilitate execution of the various functionsdescribed herein relating to the system 150, including encodingcomponent 164, learning component 166, and measuring component 168. Theencoding component 164 can be operatively coupled to both a processor160 and quantum processor 162. The quantum processor 162 can encodebased on instructions that are formed and administrated to the quantumprocessor 162 by a processor 160. The processor 160 can receive quantummeasurements, perform Bayesian learning, determine experimentalparameters, update probability/prior parameters, and translate them intoinstructions for the quantum processor 162.

As described with FIGS. 2-11 below, system 150 can generate aneigenvalue pair 145 from two quantum states 130 in accordance with oneor more embodiments described herein. One having skill in the relevantart(s), given the description herein, would appreciate that, asdiscussed herein, without departing from the spirit of one or moreembodiments described herein, eigenvalue can broadly refer to aneigenvector, an eigenvalue, as well as an eigenpair of these components.

In one or more embodiments, quantum state 130 can be any state of aquantized system that is represented by quantum numbers, wherein thequantum numbers can be numbers that describe the values of conservedquantities in a quantum system. The quantum state 130 can be employedwith technologies such as, but not limited to, quantum processingtechnologies, quantum circuit technologies, quantum computing designtechnologies, artificial intelligence technologies, machine learningtechnologies, search engine technologies, image recognitiontechnologies, speech recognition technologies, model reductiontechnologies, iterative linear solver technologies, data miningtechnologies, healthcare technologies, pharmaceutical technologies,biotechnology technologies, finance technologies, chemistrytechnologies, material discovery technologies, vibration analysistechnologies, geological technologies, aviation technologies, and/orother technologies. As a quantum state 130 can be any state of aquantized system that is represented by quantum numbers, wherein thequantum numbers can be numbers that describe the values of conservedquantities in a quantum system, the quantum state 130 can representproperties of systems in these technologies.

In one or more embodiments, quantum state 130 can comprise an associatedexpectation value. The expectation value can be a statistical mean ofmeasured values of an observable property, which can be the result of alinear operator operating on a Hilbert space. The encoding component 164can encode an expectation value in the phase of a quantum state 130. Theencoding component 164 can receive an at least partial quantum state andencode the expectation value as a phase. The phase can be encoded bysuccessive iterations of a circuit that effectively multiply a quantumstate 130 by a complex exponential that includes an amplitude. Theencoding component 164 can encode the expectation value based on anamplitude of the expectation value. The encoding component 164 canexecute on a quantum processor 162. The encoding component 164 canencode a spectrum or an eigenvalue as a phase which can be inferredutilizing the processor 160.

The learning component 166 can utilize stochastic inference to determinean eigenvalue pair 145 corresponding to the expectation value.Stochastic inference can comprise Bayesian learning. The learningcomponent can utilize a quantum processor 162 and a processor 160.

Stochastic inference can include updating the posterior probabilityportion of Bayes' law based on obtaining at least one updated sample ina Bayesian learning process. Exploiting Bayes' law in this fashion, ameasuring component 168 can measure the at least one ancilla qubit toprovide a probabilistic output that reveals information about aneigenvalue pair 145. The learning component can include a measuringcomponent 168 that can utilize at least one ancilla qubit to determinethe eigenvalue pair 145 corresponding to the expectation value. Anancilla qubit can be a qubit that can be used to store temporaryinformation that can be neither an input qubit nor an output qubit.

The learning component 166 can receive at least one ancilla qubit 205(e.g. by a learning component 166 comprising a quantum circuit 220). Thequantum circuit 220 can include a Pauli rotation that depends on acontrol parameter θ. The control parameter θ can depend on the prior,including the approximation parameter, μ. The control parameter m of thecircuit can determine an amount of controlled unitary action that can beutilized by an encoding component 164. The control parameter m can bedetermined by the prior, including the variance, σ.

FIG. 2A illustrates an example approach that can use circuit 202 toselect a control parameter m that can be used by encoding component 164to encode an expectation value in the phase of a quantum state 130, inaccordance with one or more embodiments. Repetitive description of likeelements employed in other embodiments described herein is omitted forsake of brevity.

In one or more embodiments, the intuition for control parameter m can bethe same as in the full phase estimation algorithm. One having skill inthe relevant art(s) given the description herein, would appreciate thatcircuit 202 can be used to, peel away m copies of the eigenvalue. Inanother use of circuit 202, using the full algorithm, control parameterm can run through powers of two, and select a value for controlparameter m that can maximize the information gain at each step.

Considering the above in greater detail, circuit 202 can, in one or moreembodiments, use formulas 203 for points 260 A-E, to determine controlparameter m, in accordance with one or more embodiments. in an exampleof this process, point 260 D corresponds to a ‘phase kick-back’ that canleave the eigenvector register unchanged, but change the phase of thecontrol qubit. Additional discussion of the encoding process of encodingcomponent 164 is included with a discussion of different approaches tomeasuring, discussed below.

FIG. 2B illustrates a block diagram of an example, non-limiting system200 that shows the interaction of the encoding component 164 andlearning component 166 in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements employed inother embodiments described herein is omitted for sake of brevity.

The quantum circuit 220 encodes at least one ancilla qubit 205 qubitinto a quantum state for measuring 222 that can be received by theencoding component 164. The quantum circuit 220 encodes at least oneancilla qubit 205 qubit into a quantum state for measuring 222 that canbe received by the measuring component 168. The quantum circuit 220 ofthe learning component can be executed on a quantum circuit operativelyconnected to a quantum processor 162.

The measuring component 168 can produce an output comprising aprobabilistic measurement. The probabilistic measurement can be based onat least one ancilla qubit. The learning component 166 can executepartially on a quantum processor 162 and partially on the processor 160;the measuring component 168 can interface between a quantum portion,utilizing the quantum processor 162, and a classical portion, utilizingthe processor 160.

The measuring component 168 can learn/approximate the eigenvalue pair145 by Bayesian learning, which can include Bayesian inference.Exploiting Bayes' law, the probability of a particular phase 230 (φ),given outcome (E), a variance, σ, and a mean of the initial priorfunction. For example, Bayesian analysis can begin with a prior over ϕ,namely P₀(ϕ) and can proceed by the formulas below:

${P\left( {0{❘{\phi;\theta;m}}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P\left( {1{❘{\phi;\theta;m}}} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P_{i + 1}\left( {\phi{❘{E_{i + 1};{\theta_{i;}m_{i}}}}} \right)} = \frac{{P\left( {E_{i + 1}{❘{{\phi;\theta_{i}},m_{i}}}} \right)}{P_{i}(\phi)}}{N}$

P(φ) can be updated each sample by the update component 504. Themeasuring component 168 can send its output to an update component. Theupdate component can receive probabilistic output from the measuringcomponent 168 and utilize that probabilistic output to provide a newestimate of the eigenvalue pair 145 to the measuring component 168. Themeasuring component 168 can include a classical statistical model basedon a sampling filter, which can filter based on stochastic inference.The measuring component 168 can perform a quantum measurement on theoutput of the encoding component 164. Based on a classical value outputof the quantum measurement, successive iterations can produce aclassical output 240.

The Quantum Phase Estimation method can employ a circuit depth that istechnologically unfeasible with near term quantum computing. Tocircumvent this, a portion of the circuit depth is replaced with aclassical statistical model based upon a sampling filter. Inpreparation, a quantum measurement is performed on the last stage of theencoding circuit 300, as shown in FIG. 3 , and the measurement is donein a certain basis, |k>. After the quantum measurement is performedgiving a classical value, iterations the realizations of the circuit toproduce an ensemble.

One or more embodiments can perform amplitude estimation using differentapproaches. In one approach to amplitude estimation that saturates theHeisenberg limit is to use Quantum Phase Estimation. Quantum PhaseEstimation can employ entanglement (via the inverse Quantum FourierTransform) between repeated applications of U to extract the phase. Aniterative version of Quantum Phase Estimation exists that can usefeed-forward measurements instead of entanglement.

In one approach to using phase estimation to do amplitude estimation,first the amplitude can be encoded into the phase, second, an operatorcan be defined for which eigenvectors are known, e.g., because phaseestimation relies on its preparation. In some circumstances, the U fromamplitude estimation can be arbitrary, and neither eigenvalue pairs 145nor eigenvectors are known a priori.

In one approach to defining the operator, a new rotation operator, S(related to U) can be built that, by design, can leave a certainsubspace invariant. In this approach, because S can be a two-dimensionalrotation the form of the two eigenvalue pairs 145 is: e^(±iϕ).Furthermore, because in some circumstances there are only two eigenvaluepairs 145 which differ only with respect to the signs of their phases,as long as the initial state is prepared within the subspace, “both”phases can be determined in one run of full quantum phase estimation.One having skill in the relevant art(s), given the disclosure herein,would appreciate that one or more embodiments can use this dualeigenvalue property in a Bayesian iterative context, e.g., without usingan iterative algorithm that can require projection onto one of theeigenvectors.

In an example below, the following operator can have the above describedbeneficial properties, with a rotation by ϕ=|<ψ|U|ψ>| that operates, inthis example, only in a two-dimensional subspace (spanned by |ψ₀> and|ψ₁>):

S=S ₀ ,S ₁

S ₀=∥−2|ψ₀><ψ₀|

S ₁=∥−2|ψ₁><ψ₁|=∥−2U|ψ ₀><ψ₀ |U ^(†)

where S_(x)|ψ>, x∈{0,1} is a sign flip of the component of |ψ> in thedirection of |ψ_(x)>. Each S_(x) can operate in a two dimensionalsubspace spanned by the state being acted upon and |ψ_(x)>. Therefore,if the process is started in the sub-space spanned by |ψ₀> and |ψ₁>, theresults can be determined by remaining in the sub-space.

In one or more embodiments, a sign flip of the component in thedirection of a vector can be a reflection around the line perpendicularto the vector, e.g., one or more embodiments can evaluate reflectionsabout axes. In an example implementation, two reflections around thesame axes can return a vector to where it began, and two reflectionsabout two non-coinciding axes can rotate the vector by exactly twice theangle between the two axes. This can be illustrated by how a reflectionabout the second axis can be seen as rotation to the first axis,followed by reflection about the second axis, picking up a rotation oftwice the angle between the axis followed by a rotation of the sameangle as the first. Based at least on these reflections, the angle ofseparation can be transferred between two vectors (this being related tothe inner-product between the two vectors) into the angle of rotation ofany vector in the plane. In one or more embodiments, S can be a (real)rotation by an angle ϕ=2 cos⁻¹(|

ψ₀|ψ₁

|), therefore |

ψ₀|ψ₁

|=|cos(ϕ/2)|.

In one or more embodiments, sign flips along an arbitrary vector can bewritten as a sign flip

along the zero vector |0>, by rotating to the zero vector, performingthe sign flip

and rotating back. Therefore, if the construction of |ψ₀> is specifiedas the unitary V acting on |0>, then:

S ₀ =V

V ^(†)

S ₁ =UV

V ^(†) U ^(†)

S=V

V ^(†) UV

V ^(†) U ^(†)

Thus, full phase estimation of S operating on |0₀> can reveal either ϕor −ϕ and therefore can result in the desired amplitude.

In some circumstances a ‘phase kick-back’ can leave the eigenvectorregister unchanged but can be seen as changing the phase of the controlqubit.

Therefore:

${P\left( {0{❘{\phi;\theta;m}}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P\left( {1{❘{\phi;\theta;m}}} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$

The measuring component 168 can produce an accurate measurement withoutmany samples by measuring intelligently, e.g., in a strategic way. Thisprobabilistic measurement by the measuring component 168 can utilizeBayesian learning and stochastic inference upon the quantum results. Themeasuring component 168 can perform these tasks by updating a priordistribution using Bayes' law.

The eigenvalue pair 145 can be a value corresponding to the expectationvalue in the phase of the quantum state 130. The eigenvalue pair 145 canbe a value of an observable physical quality of a linear operatoroperating on a Hilbert space. A Hilbert space is a vector space that istopologically closed under finite vector addition and scalarmultiplication with an inner product that allows properties like lengthand angle to be measured. For any observable physical quantity, thereexists a linear operator, which can be a Hermitian operator, acting onthe physical environment, which can be a Hilbert space. When we make ameasurement of property in the environment, we obtain one of theeigenvalues of the operator. Therefore, any observable physical quantitycan be modeled as the eigenvalue of a linear operator operating on aHilbert space.

The encoding component 164 can receive a quantum state 130. The encodingcomponent 164 can encode a spectrum or an eigenvalue as a phase 230,which can then be inferred classically, by a processor 160. Theclassical inference can be performed by the measuring component 168 byexploiting Bayes' law, as the probability of a particular phase 230 (φ),given outcome (E), a variance, σ, and a mean of the initial priorfunction. For example, Bayesian analysis can begin with a prior over ϕ,namely P₀(ϕ) and can proceed by the formulas below:

${P\left( {0{❘{\phi;\theta;m}}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P\left( {1{❘{\phi;\theta;m}}} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P_{i + 1}\left( {\phi{❘{E_{i + 1};{\theta_{i;}m_{i}}}}} \right)} = \frac{{P\left( {E_{i + 1}{❘{{\phi;\theta_{i}},m_{i}}}} \right)}{P_{i}(\phi)}}{N}$

P(φ) can be updated each sample by the update component 504.

The encoding component can utilize a quantum processor 162. The encodingcomponent can send the quantum state 130 to an encoding circuit 210 toencode the quantum state 130 as a phase 230 with an encoding circuit300, as shown in FIG. 3 . The encoding circuit 210 can receive thequantum state 130 and input from the learning component 166. Utilizingthe quantum state 130 and input from the learning component 166, theencoding circuit 210 can output a phase 230. This encoding can beperformed with an encoding circuit 300. The encoding circuit 300 can bethought of as repeated iterations of S, where S=S₀S₁, wherein S₀=V

V^(†)=I−2|ψ₀><ψ₀| and S₁=UV

V^(†)U^(†)=I−2U|ψ₀><ψ₀|U^(†). V represents a circuit preparation gate,V^(†) is the conjugate of V, U represents a unitary operator to infer aspectrum, U^(†) is the conjugate of U, and

=I−2|0><0|. Iterations of S lead to encoding the expectation value as aphase, φ(λ), wherein φ(λ)=2 arccos(<ψ(λ)|H|ψ(λ)>). This can encode anexpectation value of the quantum state 130, |0>|ψ> ase^(−i|<ψ(λ)|H|ψ(λ}>|)|0>|ψ>, wherein |<ψ(λ)|H|ψ(λ}>| can be an amplitudeof the expectation value. The learning component 166 receives at leastone ancilla qubit 205 and sends the at least one ancilla qubit 205 to aquantum circuit 220. In one or more embodiments, quantum circuit 220 candepend on control parameters m and θ which in turn can depend on theprobability/prior parameters μ and σ.

The approximation parameter μ can be set by a Bayesian probabilitydistribution with a tuning parameter, based on active learning (γ). Inone or more parameters, active learning can be applied to select newcontrol parameters, potentially revealing better quality information atevery Bayesian update step, as discussed above. According to certainembodiments, γ∈[0,1]. The tuning parameter can enable a gradual steeringbetween a quantum estimation and a classical estimation. In someembodiments, a quantum estimation can be used when γ=0 and a classicalestimation can be used when γ=1.

Steering between a quantum estimation and a classical estimation by theencoding component 164 using a tuning parameter, can incorporate some ofthe lower costs of both quantum and classical algorithms, as shown inFIGS. 12A and 12B. It should be noted that the eigenvector-collapse freemethod of amplitude estimation described in one or more embodimentsherein, can be a subroutine of a determination of VQE, which can conferbenefits in some circumstances, with these benefits including shorterdepth circuits and fewer samples being required, as compared to naivesampling. In VQE the goal is to produce an approximation to theground-state and calculate energy (the lowest possible) by varyingvariational parameters 2.

This is made possible by the variational principle, which states thatenergy of an arbitrary state is always greater than the ground state.Our amplitude estimation method can be used as a subroutine in VQE tocalculate the energy level of the current quantum state, e.g., startingwith the ansatz and refining.

The samples which can be employed to achieve a relatively low error Eare very high for VQE 1210. However, even at relatively low errors,γ-QPE 1220 compares quite well with QPE 1230, while not requiring thehigh fault tolerance of QPE. The circuit depth for a given number ofprocessed qubits goes up very quickly for QPE 1240. However, even atrelatively high amounts of processing, γ-QPE 1260 compares favorablywith VQE 1250. Quantum algorithms can be used for the solution of linearsystems of equations, factorization and eigenvalue decomposition, forexample. However, a problem for realizing the potential of some quantumalgorithms is that some quantum algorithms can scale proportionally to

(1/ε), wherein

denotes a function of the asymptotic upper bound approaching infinity,to obtain, a level of precision, ε. However, classical algorithms canhave a much greater computational cost in terms of sample countsrelative to quantum algorithms to form a proficient variational form forthe representation of an eigenstate. The computational cost for aclassical algorithm can grow proportional to

(1/ε²), however, each realization can necessitate a repeat preparationof the state. By first encoding the expectation value in a quantummanner and preparing it for a classical inference, a substantial speedup can be achieved.

Quantum Phase Estimation (QPE) and Variational Quantum Eigen (VQE)solver are instrumental for the computation of eigenvalues. QuantumPhase Estimation can offer an exponential advantage over classicalcomputation, in simulation of the evolution of the state of n qubits bymatrix multiplication. The algorithm can be used in applications such asthe solution of linear systems of equations, factorization, eigenvaluedecomposition, etc. A key issue for realizing the potential of QPE isthat the circuit depth scales like

(1/ϵ) in order to attain ϵ precision. A popular alternative is the VQEalgorithm that replaces the long coherence of QPE with multiple shortercoherence time expectation realizations. The drawbacks of this algorithmforming a proficient variational form for the representation of theeigenstate, and the cost of computation in terms of sample counts (whichgrow like

(1/ϵ²) for ϵ error, while repeated preparation of the state for eachiteration can be employed for each realization).

A variational quantum phase estimation approach is considered thatcombines ideas associated with rejection filtering and neural networkinference. The present embodiments offer almost continuous steeringbetween VQE and QPE, based upon the availability of computationalresources. For this reason, the algorithm is termed herein γ-QPE. Inaddition to the benefits described above, merits of using a hybridquantum classical approach include that the hybrid approach can utilizethe strength of each domain to obtain the best tradeoff between theerror E, circuit depth and the computational complexity.

The quantum circuit 220 encodes the at least one ancilla qubit 205 intoa quantum state for measuring 222 and provides input to the encodingcircuit 210. The quantum state for measuring 222 can be sent to themeasuring component 168, which can measure the ancilla qubit 205 toprovide a classical output 240, which can be a probabilistic output andcan reveal information about an eigenvalue pair 145. The quantum statefor measuring 222 can be a complex valued probability function thatrepresents a probability or probability density for a given value.

FIG. 3 illustrates an example, non-limiting encoding circuit 300 thatshows an encoding component in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements employed inother embodiments described herein is omitted for sake of brevity.

The encoding circuit 300 can receive a quantum state 130. The encodingcircuit 300 passes the eigenstate to 304, which can be a conjugate of aunitary operator, P. P can be any unitary operator that can generateinformation that can be employed by the encoding circuit 300 to generatean inference of a spectrum. P can be a Hamiltonian. The output 305 ofthe conjugate of the unitary operator 304 goes to the conjugate of thecircuit preparation circuit 306. The output of the circuit preparationcircuit 306 goes to the first order projection operator 308. By way ofexample, the first order projection operator 308, II, can be representedas Π=I−2|0><0|, wherein I is the identity matrix. The quantum state formeasuring 222 based on the at least one ancilla qubit 205 can be passedto both the conjugate of the unitary operator 304 and to a Hadamard gate302. The Hadamard gate 302 can receive a single qubit and bring it to astate superposition where its output has an equal probability of beingmeasured classically as either 0 or 1. The Hadamard gate can berepresented as a unitary matrix that is a combination of two rotations,180 degrees about the Z-axis and 90 degrees about the Y-axis. The output303 of the Hadamard gate 302 can be also passed to the first orderprojection operator 308, II, can be represented as Π=I−2|0><0|, whereinI is the identity matrix. The first order projection operator 308 can bea vacuum projection operator. The output 309 of the first orderprojection operator 308 can be passed to both another Hadamard gate 302and to the circuit preparation circuit 310. As shown at the upperportion of FIG. 3 , the output of the Hadamard gate 302 can be passed toboth another Hadamard gate 302 and to a unitary operator 312. The output303 of the Hadamard gate 302 can be passed to another first orderprojection operator 308, which can be represented as Π=I−2|0><0|. Asshown at the bottom portion of FIG. 3 , the output 309 of the firstorder projection operator 308 goes to a circuit preparation circuit 310.The output 311 of the circuit preparation circuit 310 goes to theunitary operator 312. The output 313 of the unitary operator 312 can bepassed to the conjugate of the circuit preparation circuit 306. Like theoutput 303 of the Hadamard gate 302, the output 307 of the conjugate ofthe circuit preparation circuit 306 can be passed to the first orderprojection operator 308.

The output 303 of the first order projection operator 308 can be passedto a Hadamard gate 302 and to a circuit preparation circuit 310. Thisencoding circuit 300 can be iterated as necessary to reduce error inencoding a phase.

FIG. 4 illustrates an example, non-limiting geometric representation ofthe example, non-limiting circuit of FIG. 3 . The encoding circuit 300can encode an absolute expectation value as a phase by a rotation S,where S=S₀S₁. The phase can be measured. The encoding circuit 300 canutilize partial projection operators. Repetitive description of likeelements employed in other embodiments described herein is omitted forsake of brevity. As shown on the Bloch sphere 400, geometrically, thecircuit of FIG. 3 can be thought of as repeated iterations of S, whereS=S₀S₁, wherein S0=VΠV^(†)=I−2|ψ₀><ψ₀| and S₁=UVΠV^(†)U^(†)=I−2U|Ψ|ψ₀^(†). S can be a rotation by an angle ϕ=2 arccos

ψ(λ)|H|ψ(λ)

. S₀ and S₁ can coincide with the geometrical axis shown in FIG. 4 andcan be a geometrical representation and rotation of subset projectionoperators. The first order projection operator 308 can be a vacuumoperator which can be Π=I−2|0><0|. Iterations of S lead to encoding theexpectation value as a phase, φ(λ) wherein φ(λ)=2 arccos(<ψ(λ)|H|ψ(λ)>).

FIG. 5 illustrates a block diagram of an example, non-limiting systemthat can generate an eigenvalue from a quantum state in accordance withone or more embodiments described herein. Repetitive description of likeelements employed in other embodiments described herein is omitted forsake of brevity.

The quantum state 130 can be received by the encoding component 164. Theencoding component 164 can encode the quantum state 130 as a phase 230based on an amplitude of the expectation value of the quantum state 130.The phase 230 can be passed to the learning component 166. The encodingcomponent can be executed by a quantum processor 162.

The learning component 166 can receive an at least one ancilla qubit205. The at least one ancilla qubit 205 can be received by the quantumcircuit 220 of the learning component 166. The quantum circuit 220 canbe a Pauli rotation that includes a variance, σ, and a mean of theinitial prior function, μ. The quantum circuit 220 encodes at least oneancilla qubit into a quantum state for measuring 222 that can bereceived by the encoding component 164. The quantum circuit 220 of thelearning component 166 can be executed on a quantum circuit operativelyconnected to a quantum processor 162.

The learning component 166 can comprise a measuring component 168. Themeasuring component 168 can receive a quantum state for measuring 222from the quantum circuit 220. The probability of a given outcome, whichcan be a value corresponding to the expectation value of the quantumstate 130, can be calculated explicitly on by the measuring component168 according to the equation, below,

${P\left( {0{❘{\phi;\theta;m}}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P\left( {1{❘{\phi;\theta;m}}} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$

thereby measuring the eigenvalue probabilistically based on the at leastone ancilla qubit. The measuring component can receive the phase 230from the encoding component 164, represented as φ(λ)=2arccos(<ψ(λ)|H|ψ(λ)>).

The encoding component 164 can receive a quantum state 130. The encodingcomponent 164 can encode a spectrum or an eigenvalue as a phase 230,which can then be inferred classically, by a processor 160. Theclassical inference can be performed by the measuring component 168 byexploiting Bayes' law, as the probability of a particular phase 230 (φ),given outcome (E), a variance, σ, and a mean of the initial priorfunction. For example, Bayesian analysis can begin with a prior over ϕ,namely P₀(ϕ) and can proceed by the formulas below:

${P\left( {0{❘{\phi;\theta;m}}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P\left( {1{❘{\phi;\theta;m}}} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}$${P_{i + 1}\left( {\phi{❘{E_{i + 1};{\theta_{i;}m_{i}}}}} \right)} = \frac{{P\left( {E_{i + 1}{❘{{\phi;\theta_{i}},m_{i}}}} \right)}{P_{i}(\phi)}}{N}$

P(φ) can be updated each sample by the update component 504.

The measuring component 168 can send its output to an update component504. The measuring component 168 can execute on both the quantumprocessor 162 and processor 160. The update component 504 can receivethe probabilistic output 508 of the measuring component 168 and utilizethat probabilistic output 508 to provide a new upper bound 506 of theeigenvalue pair 145 to the measuring component 168. |<ψ(λ)|H|ψ(λ}>| isthe absolute value of the expectation value, which can be encoded as thephase of a quantum state, |0>|ψ>, quantum state 130 can be transformedby the encoding component 164 into e^(−i|<ψ(λ)|H|ψ(λ)>|)|0>|ψ>. Theexpression <ψ|H|ψ> can also be expressed as:

$\begin{matrix}{\sum\limits_{{\lambda 1},{{\lambda 2} \in {{Spec}(H)}}}{< {\psi{❘{\psi_{\lambda_{1}} > < {\psi_{\lambda_{1}}{❘H❘}\psi_{\lambda_{2}}} > < {\psi_{\lambda_{2}}{❘{\psi >}}}}}}}} & {{Equation}(4)}\end{matrix}$

which is equivalent to

$\begin{matrix}{\sum\limits_{\lambda \in {{Spec}(H)}}{\lambda{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} & \left( {{Equation}(5)} \right.\end{matrix}$

Equation (4) and Equation (5) can be always greater than or equal to

$\begin{matrix}{{\sum\limits_{\lambda \in {{Spec}(H)}}{E_{0}{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} = {E_{0}.}} & {{Equation}(6)}\end{matrix}$

wherein E₀ is the minimum eigenvalue of H. VQE, employs this variationalprinciple to search for the lowest eigenvalue by adjusting thevariational parameters λ. At each stage of adjustment the expectationvalue needs to be calculated. The learning component 166 can utilizeboth stochastic inference and Bayesian inference to calculate theexpectation value. After obtaining a sample, the posterior probability(P(φ|E, μ, σ)) of the Bayesian learning equation can be updated with anew variance (σ) and mean of the initial function (μ). Further, as σ andμ are updated by the update component 504, the measuring component 168can be independent of a given initial state.

FIG. 6 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can generate an eigenvalue from aquantum state in accordance with one or more embodiments describedherein. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity. As shown bythe diagram 600, at 602, a quantum state 130 can be encoded (e.g., viathe encoding component 164). The quantum state 130 can be associatedwith technologies such as, but not limited to, quantum processingtechnologies, quantum circuit technologies, quantum computing designtechnologies, artificial intelligence technologies, machine learningtechnologies, search engine technologies, image recognitiontechnologies, speech recognition technologies, model reductiontechnologies, iterative linear solver technologies, data miningtechnologies, healthcare technologies, pharmaceutical technologies,biotechnology technologies, finance technologies, chemistrytechnologies, material discovery technologies, vibration analysistechnologies, geological technologies, aviation technologies, and/orother technologies. The quantum state 130 can be encoded as a phase 230.The encoding component can receive a quantum state for measuring 222from a learning component 166. The encoding component can pass the phase230 to the learning component 166. The encoding component 164 can aquantum state 130, which can be an at least partial eigenstate. Theencoding component can send the quantum state 130 to an encoding circuit210. The encoding circuit 210 can receive the quantum state 130 andinput from the learning component 166. Utilizing the quantum state 130and input from the learning component 166, the encoding circuit 210 canoutput a phase 230.

At 604, in one or more embodiments, learning component 166 can learn aneigenvalue corresponding to the expectation value by stochasticinference. Learning component 166 can receive a phase 230 from theencoding component 164. The learning component can pass a quantum statefor measuring 222 based on the at least one ancilla qubit 205 to theencoding component 164. In one or more embodiments, quantum circuit 220can depend on control parameters m and θ which in turn can depend on theprobability/prior parameters μ and σ. The quantum circuit 220 encodesthe at least one ancilla qubit 205 into a quantum state for measuring222 and provides input to the encoding circuit 210. The quantum statefor measuring 222 can be sent to the measuring component 168, which canmeasure to provide a classical output 240, which can be an eigenvaluepair 145. In one or more embodiments, learning component 166 can learnan eigenvalue corresponding to the expectation value by stochasticinference by a process that comprises measuring the quantum state 606 toproduce an output comprising a probabilistic measurement.

FIG. 7 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can encode a quantum state 130 as aphase 230 in accordance with one or more embodiments described herein.Repetitive description of like elements employed in other embodimentsdescribed herein is omitted for sake of brevity. As shown in the blockdiagram 700, at 702, a quantum state can be received (e.g., by anencoding component 164 comprising an encoding circuit 300). The encodingcircuit 300 passes the eigenstate to 304, which can be a conjugate of aunitary operator, P. P can be any unitary operator that can generateinformation that can be employed by the encoding circuit 300 to generatean inference of a spectrum. P can be a Hamiltonian. The output 305 ofthe conjugate of the unitary operator 304 goes to the conjugate of thecircuit preparation circuit 306. The output of the circuit preparationcircuit 306 goes to the first order projection operator 308. By way ofexample, the first order projection operator 308, II, can be representedas Π=I−2|0><0|, wherein I is the identity matrix.

At 704, a quantum state for measuring 222 based on an at least oneancilla qubit 205 can be received (e.g., by an encoding component 164comprising an encoding circuit 300). The quantum state for measuring 222based on the at least one ancilla qubit 205 can be passed to both theconjugate of the unitary operator 304 and to a Hadamard gate 302. TheHadamard gate 302 can receive a single qubit and bring it to a statesuperposition where its output has an equal probability of beingmeasured classically as either 0 or 1. The Hadamard gate can berepresented as a unitary matrix that is a combination of two rotations,180 degrees about the Z-axis and 90 degrees about the Y-axis. The output303 of the Hadamard gate 302 can be also passed to the first orderprojection operator 308, Π, can be represented as Π=I−2|0><0|, wherein Iis the identity matrix.

The output of the first order projection operator 308 can be passed toboth another Hadamard gate 302 and to the circuit preparation circuit310. As shown at the upper portion of FIG. 3 , the output of theHadamard gate 302 can be passed to both another Hadamard gate 302 and toa unitary operator 312. The output of the Hadamard gate 302 can bepassed to another first order projection operator 308, which can berepresented as Π=I−2|0><0|. As shown at the bottom portion of FIG. 3 ,the output 309 of the first order projection operator 308 goes to acircuit preparation circuit 310. The output 311 of the circuitpreparation circuit 310 goes to the unitary operator 312. The output 313of the unitary operator 312 can be passed to the conjugate of thecircuit preparation circuit 306. Like the output 303 of the Hadamardgate 302, the output 307 of the conjugate of the circuit preparationcircuit 306 can be passed to the first order projection operator 308.

The output 303 of the first order projection operator 308 can be passedto a Hadamard gate 302 and to a circuit preparation circuit 310. Thisencoding circuit 300 can be iterated as necessary to reduce error inencoding a phase.

At 706, the quantum state can be encoded as a phase (e.g., by anencoding component 164 comprising an encoding circuit 300). The encodingcomponent 164 can execute on a quantum processor 162. Iterations of theencoding circuit 300, can be represented mathematically as repeatediterations of S, where S=S₀S₁, wherein S₀=VΠV^(†)=I−2|ψ₀><ψ₀| andS₁=UVΠV^(†)U^(†)=I−2U|ψ₀><ψ₀|U^(†). Iterations of S lead to encoding theexpectation value as a phase, φ(λ), wherein φ(λ)=2arccos(<ψ(λ)|H|ψ(λ)>). This can encode an expectation value of thequantum state 130, |0>|ψ> as e^(−i|<ψ(λ)|H|ψ(λ}>|)|0>|ψ>, wherein|<ψ(λ)|H|ψ(λ)>I can be an amplitude of the expectation value.

FIG. 8 illustrates a flow diagram 800 of a computer-implemented methodthat can learn an eigenvalue corresponding to an expectation value basedon a quantum state in accordance with one or more embodiments describedherein. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity. At 802, thelearning component 166 can receive an at least one ancilla qubit 205(e.g. by a learning component 166 comprising a quantum circuit 220). Inone or more embodiments, quantum circuit 220 can depend on controlparameters m and θ which in turn can depend on the probability/priorparameters μ and σ. The quantum circuit 220 encodes at least one ancillaqubit 205 qubit into a quantum state for measuring 222 that can bereceived by the encoding component 164. The quantum circuit 220 encodesat least one ancilla qubit 205 qubit into a quantum state for measuring222 that can be received by the measuring component 168. The quantumcircuit 220 of the learning component can be executed on a quantumcircuit operatively connected to a quantum processor 162.

At 804, the eigenvalue corresponding to the expectation value can bedetermined by measuring an ancilla qubit (e.g., via a measuringcomponent 168). The learning component 166 can comprise a measuringcomponent 168. The measuring component 168 can receive a quantum statefor measuring 222 from the quantum circuit 220. The probability of agiven eigenvalue pair 145, which can be a value corresponding to theexpectation value of the quantum state 130, can be calculated explicitlyon by the measuring component 168 according to the equations below,

${{P\left( {{0❘\phi};\theta;m} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P\left( {{1❘\phi};\theta;m} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P_{i + 1}\left( {{{\phi ❘E_{i + 1}};\theta_{i}},m_{i}} \right)} = \frac{{P\left( {{{E_{i + 1}❘\phi};\theta_{i}},m_{i}} \right)}{P_{i}(\phi)}}{N}}$

thereby measuring the eigenvalue probabilistically based on the at leastone ancilla qubit. The measuring component can receive the phase 230from the encoding component 164, represented as φ(λ)=2arccos(<ψ(λ)|H|ψ(λ)>).

The classical inference can be performed by the measuring component 168by exploiting Bayes' law, as the probability of a particular phase 230(φ), given outcome (E), a variance, σ, and a mean of the initial priorfunction. For example, Bayesian analysis can begin with a prior over ϕ,namely P₀(ϕ) and can proceed by the formulas below:

${{P\left( {{0❘\phi},{\theta;m}} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P\left( {{1❘\phi};\theta;m} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P_{i + 1}\left( {{{\phi ❘E_{i + 1}};\theta_{i}},m_{i}} \right)} = \frac{{P\left( {{{E_{i + 1}❘\phi};\theta_{i}},m_{i}} \right)}{P_{i}(\phi)}}{N}}$

P(ϕ) can be updated each sample by the update component 504.

At 806, the upper bound in the measuring component 168 can be updated(e.g., via an update component 504). The measuring component 168 canmeasure the eigenvalue pair 145 by Bayesian learning, which can includeBayesian inference. The classical inference can be performed by themeasuring component 168 by exploiting Bayes' law, as the probability ofa particular phase 230 (φ), given outcome (E), a variance, σ, and a meanof the initial prior function. For example, Bayesian analysis can beginwith a prior over ϕ, namely P₀(ϕ) and can proceed by the formulas below:

${{P\left( {{0❘\phi};\theta;m} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P\left( {{1❘\phi};\theta;m} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P_{i + 1}\left( {{{\phi ❘E_{i + 1}};\theta_{i}},m_{i}} \right)} = \frac{{P\left( {{{E_{i + 1}❘\phi};\theta_{i}},m_{i}} \right)}{P_{i}(\phi)}}{N}}$

P(ϕ) can be updated each sample by the update component 504.

The measuring component 168 can send its output to an update component504. The measuring component 168 can execute on both the quantumprocessor 162 and processor 160. The update component 504 can receivethe probabilistic output 508 of the measuring component 168 and utilizethat probabilistic output 508 to provide a new upper bound 506 of theeigenvalue pair 145 to the measuring component 168. |<ψ(λ)|H|ψ(λ}>| canbe an amplitude on which the phase 230 can be encoded, as the quantumstate 130, |0>|ψ>, can be encoded by the encoding component 164 ase^(−i|<ψ(λ)|H|ψ(λ}>|)|0>|ψ>. The expression <ψ|H|ψ> can also beexpressed as:

$\begin{matrix}{\sum\limits_{\lambda_{1},{\lambda_{2} \in {{Spec}(H)}}}{\left\langle {\psi ❘\psi_{{\lambda}_{1}}} \right\rangle\left\langle {\psi_{\lambda_{1}}{❘H❘}\psi_{\lambda_{2}}} \right\rangle\left\langle {\psi_{\lambda_{2}}❘\psi} \right\rangle}} & {{Equation}(4)}\end{matrix}$

which is equivalent to

$\begin{matrix}{\sum\limits_{\lambda \in {{Spec}(H)}}{\lambda{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} & \left( {{Equation}(5)} \right.\end{matrix}$

Equation (4) and Equation (5) can be always greater than or equal to

$\begin{matrix}{{\sum\limits_{\lambda \in {{Spec}(H)}}{E_{0}{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} = {E_{0}.}} & {{Equation}(6)}\end{matrix}$

wherein E₀ can be representative of the quantum state 130. In this way,the learning component 166 can utilize both stochastic inference andBayesian inference. After obtaining a sample, the posterior probability(P(φ|E, μ, σ)) of the Bayesian learning equation can be updated with anew variance (σ) and mean of the initial function (μ). Further, as σ andμ are updated by the update component 504, the measuring component 168can be independent of an input state.

FIG. 9 illustrates a flow diagram of an example, non-limitingcomputer-implemented method that can generate an eigenvalue from aquantum state in accordance with one or more embodiments describedherein. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity. At 902, aquantum state 130 can be encoded as a phase. Encoding a quantum state asa phase 902 can include receiving a quantum state 904 and receiving aquantum state 906.

The quantum state 130 can be received by the encoding component 164. Theencoding component 164 can encode the quantum state 130 as a phase 230based on an amplitude of the expectation value of the quantum state 130.The phase 230 can be passed to a learning component 166. The encodingcomponent can be executed by a quantum processor 162.

At 908, an eigenvalue corresponding to the expectation value can belearned, by the system, by stochastic inference (e.g., via a learningcomponent 166). The learning component 166 can receive an at least oneancilla qubit 205. The at least one ancilla qubit 205 can be received bythe quantum circuit 220 of the learning component 166. In one or moreembodiments, quantum circuit 220 can depend on control parameters m andθ which in turn can depend on the probability/prior parameters μ and σ.The quantum circuit 220 encodes at least one ancilla qubit into aquantum state for measuring 222 that can be received by the encodingcomponent 164. The quantum circuit 220 of the learning component can beexecuted on a quantum circuit operatively connected to a quantumprocessor 162.

The learning component 166 can comprise a measuring component 168. Themeasuring component 168 can receive a quantum state for measuring 222from the quantum circuit 220. The probability of a given eigenvalue pair145, which can be a value corresponding to the expectation value of thequantum state 130, can be calculated explicitly on by the measuringcomponent 168 according to the equations below:

${{P\left( {{0❘\phi};\theta;m} \right)} = \frac{1 + {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P\left( {{1❘\phi};\theta;m} \right)} = \frac{1 - {\cos\left( {m\left( {\theta - \phi} \right)} \right)}}{2}}{{P_{i + 1}\left( {{{\phi ❘E_{i + 1}};\theta_{i}},m_{i}} \right)} = \frac{{P\left( {{{E_{i + 1}❘\phi};\theta_{i}},m_{i}} \right)}{P_{i}(\phi)}}{N}}$

thereby measuring the eigenvalue probabilistically based on the at leastone ancilla qubit. The measuring component can receive the phase 230from the encoding component 164, represented as φ(λ)=2arccos(<ψ(λ)|H|ψ(λ)>).

The measuring component 168 can measure the eigenvalue pair 145 byBayesian learning, which can include Bayesian inference. ExploitingBayes' law, the probability of a particular phase 230 (φ) with a giveneigenvalue (E), μ, and σ can be written as P(φ|E, μ, σ)=[P(E, μ, σ|φ)P(φ)]/[∫P(E, μ, σ|φ)P(φ)dφ]. P(φ) can be updated each sample by theupdate component 504.

The measuring component 168 can measure the eigenvalue pair 145 byBayesian learning, which can include Bayesian inference. ExploitingBayes' law, the probability of a particular phase 230 (φ) with a giveneigenvalue (E), μ, and σ can be written as P(φ|E, μ, σ)=[P(E, μ, σ|φ)P(φ)]/[∫P(E, μ, σ|φ) P(φ)dφ]. P(φ) can be updated each sample by theupdate component 504.

The measuring component 168 can send its output to an update component504. The measuring component 168 can execute on both the quantumprocessor 162 and processor 160. The update component 504 can receivethe probabilistic output 508 of the measuring component 168 and utilizethat probabilistic output 508 to provide a new upper bound 506 of theeigenvalue pair 145 to the measuring component 168. |<ψ(λ)|H|104 (λ)>|can be an amplitude on which the phase 230 can be encoded, as thequantum state 130, |0>ψ>, can be encoded by the encoding component 164as e^(−i|ψ(λ)|H|ψ(λ}>|)|0>|ψ>. The expression <ψ|H|ψ> can also beexpressed as:

$\begin{matrix}{\sum\limits_{\lambda_{1},{\lambda_{2} \in {{Spec}(H)}}}{\left\langle {\psi ❘\psi_{\lambda_{1}}} \right\rangle\left\langle {\psi_{\lambda_{1}}{❘H❘}\psi_{\lambda_{2}}} \right\rangle\left\langle {\psi_{\lambda_{2}}❘\psi} \right\rangle}} & {{Equation}(4)}\end{matrix}$

which is equivalent to

$\begin{matrix}{\sum\limits_{\lambda \in {{Spec}(H)}}{\lambda{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} & \left( {{Equation}(5)} \right.\end{matrix}$

Equation (4) and Equation (5) can be always greater than or equal to

$\begin{matrix}{{\sum\limits_{\lambda \in {{Spec}(H)}}{E_{0}{❘\left\langle {\psi_{\lambda}❘\psi} \right\rangle ❘}^{2}}} = {E_{0}.}} & {{Equation}(6)}\end{matrix}$

wherein E₀ can be representative of the quantum state 130. In this way,the learning component 166 can utilize both stochastic inference andBayesian inference. After obtaining a sample, the posterior probability(Pφ|E, μ, σ)) of the Bayesian learning equation can be updated with anew variance (σ) and mean of the initial function (μ). Further, as σ andμ are updated by the update component 504, the measuring component 168can be independent of a given input state.

Moreover, because at least encoding an expectation value associated witha quantum state and learning, by the system, an eigenvalue correspondingto the expectation value by stochastic inference, etc. are establishedfrom a combination of electrical and mechanical components andcircuitry, a human is unable to replicate or perform processingperformed by the processor or quantum processor (e.g., the learningcomponent, measuring component, encoding component, etc.) disclosedherein. For example, a human is unable to encode an expectation value,etc.

Moreover, because encoding, measuring, learning, updating, etc. and/orcommunication between processing components and/or an assignmentcomponent is established from a combination of electrical and mechanicalcomponents and circuitry, a human is unable to replicate or perform thesubject measuring, learning, updating, etc. and/or the subjectcommunication between processing components and/or a database component.For example, a human is unable to encode an expectation value associatedwith a quantum state as a phase on a system comprising a processoroperatively coupled to memory, etc. Moreover, a human is unable tolearn, by a system, an eigenvalue corresponding to the expectation valueby stochastic inference, etc.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 10 as well as the following discussion are intendedto provide a general description of a suitable environment in which thevarious aspects of the disclosed subject matter can be implemented. FIG.10 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity.

With reference to FIG. 10 , a suitable operating environment 1000 forimplementing various aspects of this disclosure can also include acomputer 1012. The computer 1012 can also include a processing unit1014, a system memory 1016, and a system bus 1018. The system bus 1018couples system components including, but not limited to, the systemmemory 1016 to the processing unit 1014. The processing unit 1014 can beany of various available processors. Dual microprocessors and othermultiprocessor architectures also can be employed as the processing unit1014. The system bus 1018 can be any of several types of busstructure(s) including the memory bus or memory controller, a peripheralbus or external bus, and/or a local bus using any variety of availablebus architectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and SmallComputer Systems Interface (SCSI).

The system memory 1016 can also include volatile memory 1020 andnonvolatile memory 1022. The basic input/output system (BIOS),containing the basic routines to transfer information between elementswithin the computer 1012, such as during start-up, is stored innonvolatile memory 1022. Computer 1012 can also includeremovable/non-removable, volatile/non-volatile computer storage media.FIG. 10 illustrates, for example, a disk storage 1024. Disk storage 1024can also include, but is not limited to, devices like a magnetic diskdrive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100drive, flash memory card, or memory stick. The disk storage 1024 alsocan include storage media separately or in combination with otherstorage media. To facilitate connection of the disk storage 1024 to thesystem bus 1018, a removable or non-removable interface is typicallyused, such as interface 1026. FIG. 10 also depicts software that acts asan intermediary between users and the basic computer resources describedin the suitable operating environment 1000. Such software can alsoinclude, for example, an operating system 1028. Operating system 1028,which can be stored on disk storage 1024, acts to control and allocateresources of the computer 1012.

System applications 1030 take advantage of the management of resourcesby operating system 1028 through program modules 1032 and program data1034, e.g., stored either in system memory 1016 or on disk storage 1024.It is to be appreciated that this disclosure can be implemented withvarious operating systems or combinations of operating systems. A userenters commands or information into the computer 1012 through inputdevice(s) 1036. Input devices 1036 include, but are not limited to, apointing device such as a mouse, trackball, stylus, touch pad, keyboard,microphone, joystick, game pad, satellite dish, scanner, TV tuner card,digital camera, digital video camera, web camera, and the like. Theseand other input devices connect to the processing unit 1014 through thesystem bus 1018 via interface port(s) 1038. Interface port(s) 1038include, for example, a serial port, a parallel port, a game port, and auniversal serial bus (USB). Output device(s) 1040 use some of the sametype of ports as input device(s) 1036. Thus, for example, a USB port canbe used to provide input to computer 1012, and to output informationfrom computer 1012 to an output device 1040. Output adapter 1042 isprovided to illustrate that there are some output devices 1040 likemonitors, speakers, and printers, among other output devices 1040, whichrequire special adapters. The output adapters 1042 include, by way ofillustration and not limitation, video and sound cards that provide ameans of connection between the output device 1040 and the system bus1018. It should be noted that other devices and/or systems of devicesprovide both input and output capabilities such as remote computer(s)1044.

Computer 1012 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)1044. The remote computer(s) 1044 can be a computer, a server, a router,a network PC, a workstation, a microprocessor based appliance, a peerdevice or other common network node and the like, and typically can alsoinclude many or all of the elements described relative to computer 1012.For purposes of brevity, only a memory storage device 1046 isillustrated with remote computer(s) 1044. Remote computer(s) 1044 islogically connected to computer 1012 through a network interface 1048and then physically connected via communication connection 1050. Networkinterface 1048 encompasses wire and/or wireless communication networkssuch as local-area networks (LAN), wide-area networks (WAN), cellularnetworks, etc. LAN technologies include Fiber Distributed Data Interface(FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ringand the like. WAN technologies include, but are not limited to,point-to-point links, circuit switching networks like IntegratedServices Digital Networks (ISDN) and variations thereon, packetswitching networks, and Digital Subscriber Lines (DSL). Communicationconnection(s) 1050 refers to the hardware/software employed to connectthe network interface 1048 to the system bus 1018. While communicationconnection 1050 is shown for illustrative clarity inside computer 1012,it can also be external to computer 1012. The hardware/software forconnection to the network interface 1048 can also include, for exemplarypurposes only, internal and external technologies such as, modemsincluding regular telephone grade modems, cable modems and DSL modems,ISDN adapters, and Ethernet cards.

FIG. 11 illustrates an example, non-limiting graph showing an exampleprobabilistic measurement of a phase in accordance with one or moreembodiments described herein. Repetitive description of like elementsemployed in other embodiments described herein is omitted for sake ofbrevity. The graph 1100 shows plot points 1120 of the probabilisticmeasurement that can be done by the measuring component 168.

Quantum Bayesian Amplitude Estimation: One or more embodiments describedherein can combine quantum expectation estimation with quantum Bayesianphase estimation, while avoiding problems associated with eigenvectorcollapse.

In contrast to the use of a full phase estimation or iterative phaseestimation, one or more embodiments can use a non-trivial variant of aquantum Bayesian phase estimation process. In some uses of quantumBayesian estimation, a state is initialised into an eigenvector. Incontrast, in one or more embodiments, an S rotation operator and theinitial state |ψ₀> can be used by selecting certain θ_(i) values. One ormore embodiments can avoid a problem where |ψ₀> has components in thedirection of both eigenvectors, that is, picking up both phases ±ϕ, andpotentially preventing the ability to construct a single pair oflikelihoods that is independent of the unknown eigenvector decompositionof the initial state. To avoid this result, one or more embodiments canselect θ such that the likelihood is invariant under a sign change of ϕ.In some implementations, this approach can leads to two possible choices

θ∈{0,π/2m}

These two choices allow that for any m (and higher m's can be used asprecision improves) there can be significant statisticaldistinguishability between measuring a zero versus a one.

This can result, in some circumstances, because of differences betweentwo outcome of either cos (mϕ) or sin(mϕ), with the larger of whichbeing generally greater than 1/√{square root over (2)}.

One or more embodiments can be implemented where prior knowledge of thephase can be incorporated in order to speed up convergence. For example,in some uses of quantum homology, quantum expectation estimation can beused to calculate betti numbers, these being known to be integer values.

Maximizing Information Gain: One or more embodiments can analyzeinformation gain as defined by the expected utility, where the utilityis defined as the Kullback-Liebler divergence between the prior andposterior distributions. In one or more embodiments, this approach canyield useful formulations, as well as a novel understanding of theKitaev protocol.

One or more embodiments can determine an expected utility by thedifference between the entropy of the outcome of the experimentaveraging out prior knowledge and the entropy taking into account priorknowledge. One approach to implementing this determination is to choosem and θ to maximise this expected utility. Because, in somecircumstances, possible values of θ (given m) can be restricted to two,one or more embodiments can use the following approach to define theexpected utility for these two choices of θ. For example, with θ=0:

${{EU}\left( {m_{i},{\theta_{i} = 0}} \right)} = {{H\left( {\frac{1}{2} + {\frac{\pi}{2}{{\hat{P}}_{i}^{E}(m)}}} \right)} - {\int_{- \pi}^{\pi}{{H\left( {\frac{1}{2} + \frac{\cos\left( {m\phi} \right)}{2}} \right)}{P_{i}^{E}(\phi)}d\phi}}}$

where,

${{{\hat{P}}_{i}^{E}(m)}:={\frac{1}{\pi}{\int_{- \pi}^{\pi}{{\cos\left( {m\phi} \right)}{P_{i}(\phi)}d\phi}}}},$

is the even Fourier component (the cosine component) of the prior atfrequency m and P_(i) ^(E)(ϕ): =(P_(i)(ϕ)+P_(i)(−ϕ))/2 is the even partof the prior.

In another example, θ=π/2m:

${{EU}\left( {m_{i},{\theta_{i} = {\pi/2m}}} \right)} = {{H\left( {\frac{1}{2} + {\frac{\pi}{2}{{\hat{P}}_{i}^{O}(m)}}} \right)} - {\int_{- \pi}^{\pi}{{H\left( {\frac{1}{2} + \frac{\sin\left( {m\phi} \right)}{2}} \right)}{P_{i}(\phi)}d\phi}}}$

Thus, in this example approach, an example optimal choice of θ and m isthe one that maximises the maximum of the above two expressions.

The measuring component 168 can produce an accurate measurement withoutmany samples by measuring intelligently. This probabilistic measurementcan utilize Bayesian learning and stochastic inference upon the quantumresults. This can be performed by updating a prior distribution usingBayes' law.

Based on this probabilistic distribution, the measuring component 168can start from an initial Gaussian prior with mean μ and variance σ².Bayesian inference can be seen as an update of the prior distribution toproduce the posterior distribution. As a result of the updated posteriordistribution, measuring component 168 can replace the prior distributionwith the posterior distribution. This process can be iterated for eachof the random measurements in a data set.

Bayesian inference can be based on a discrete distribution. The discretedistribution can be obtained by sampling from a discrete set of samplesbased on a prior distribution. This can be done on a classical processor160.

The measuring component 168 can measure for a given μ and σ, therebyobserving the outcome E∈{0,1} for a given number of i samples. For eachsample, the measuring component 168 can obtain a phase ϕ_(j) and assignϕ_(j) to Φ_(accept) with probability P(E|ϕ_(j),θ,M)/κ_(E), whereκ_(E)∈(0,1] is a constant constrained by P(E|ϕ_(j),θ,M)/κ_(E)≤1∀ϕ_(j),E. Thereby, the measuring component 168 can return μ=

(Φ_(accept)) and σ=√{square root over (

(Φ_(accept)) )}. As a first approximation and in order to reduce theBayes risk, the measuring component 168 can choose μ=┌1.25/σ┐.

FIGS. 12A and 12B illustrate example, non-limiting graphs showing thesamples for a given error ϵ. and circuit depth for a given number ofqubits.

As shown in FIG. 12A, the samples which can be employed to achieve arelatively low error ϵ are very high for VQE 1210. However, even atrelatively low errors, γ-QPE 1220 compares quite well with QPE 1230,while not requiring the high fault tolerance of QPE.

As shown in FIG. 12B, the circuit depth for a given number of processedqubits goes up very quickly for QPE 1240. However, even at relativelyhigh amounts of processing, γ-QPE 1260 compares favorably with VQE 1250.

While VQE and QPE scale well in different categories, the γ-QPE scalesintermodally in both categories of precision and number of gates showingthe tradeoff between precision (error) and circuit number (circuitdepth). Allowing such continues steering lets one change the scalingdepending on the tasks at hand giving an advantage in hard quantuminference tasks.

Eigenvalues can be a value of an observable physical quality of a linearoperator operating on a Hilbert space. Eigenvalues can correspond to theexpectation values of a quantum state in a Hilbert space. Classical andquantum algorithms facilitating the determination of these eigenvaluescome with competing costs.

However, classical algorithms do provide some relative benefits to purequantum algorithms Where the ansatz of a quantum algorithm can be rigidand non-adaptive, the ansatz of a classical algorithm can be relativelyflexible, bounded only by the ansatz span. Further, classical algorithmscan be less dependent upon an input state than quantum algorithms, anddo not require an extremely high fault tolerance.

While quantum algorithms have shown significant results in many areas,as the complexity of the problems increases, the rigidity and faulttolerance requirements of quantum algorithms becomes problematic.Therefore, a hybrid classical-quantum approach, such as γ-QPE, canincorporate some of the lower costs of both quantum and classicalalgorithms.

The system and/or the components of the system can be employed to usehardware and/or software to solve problems that are highly technical innature (e.g., related to encoding, measuring, machine learning) that arenot abstract and that cannot be performed as a set of mental acts by ahuman because they refer to specific machine processes. For example, ahuman cannot encode a quantum state as a phase on computer-readablemedium, as a human cannot read a computer readable medium, or learn aneigenvalue associated with a quantum state on a computer-readablemedium. Further, some of the processes, such as receiving a quantumstate and measuring an ancilla probabilistically, can be performed byspecialized computers facilitating carrying out defined tasks related tothe quantum state estimation subject area. The system and/or componentsof the system can be employed to solve problems of greater complexityand that require greater computational power that arise throughadvancements in technology. The system can provide technicalimprovements to solving for eigenvalues by improving processingefficiency, reliability, and flexibility by providing a hybridclassical-quantum approach for quantum state estimation. In this way,the system can reduce errors in quantum state estimation by providingimproved scaling and depth for a given processing cost, and therebyimprove computer functionality in solving for eigenvalues whichrepresent observables, etc.

The present invention can be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks and/or implement particular abstract datatypes. Moreover, those skilled in the art will appreciate that theinventive computer-implemented methods can be practiced with othercomputer system configurations, including single-processor ormultiprocessor computer systems, mini-computing devices, mainframecomputers, as well as computers, hand-held computing devices (e.g., PDA,phone), microprocessor-based or programmable consumer or industrialelectronics, and the like. The illustrated aspects can also be practicedin distributed computing environments in which tasks are performed byremote processing devices that are linked through a communicationsnetwork. However, some, if not all aspects of this disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

1. A system comprising: a memory that stores computer executablecomponents, and a processor that executes the computer executablecomponents stored in the memory, wherein the computer executablecomponents comprise: a learning component that utilizes stochasticinference to determine an expectation value during a first time intervalbased on an uncollapsed eigenvalue pair retrieved by an active learningprocess, wherein the first time interval is less than a second timeinterval where the uncollapsed eigenvalue pair is retrieved without useof the active learning process; and an encoding component that encodesthe expectation value associated with a quantum state. 2-3. (canceled)4. The system of claim 1, further comprising a quantum processor,wherein the encoding component encodes the expectation value as a phase,wherein the encoding component is operatively coupled to the quantumprocessor.
 5. The system of claim 1, wherein the stochastic inferencecomprises Bayesian learning of an eigenvalue corresponding to theuncollapsed eigenvalue pair.
 6. The system of claim 1, wherein theencoding component encodes the expectation value based on an amplitudeof the expectation value.
 7. The system of claim 1, wherein the learningcomponent comprises a measuring component that utilizes a first ancillaqubit to retrieve the uncollapsed eigenvalue pair.
 8. The system ofclaim 7, wherein the measuring component further utilizes a secondancilla qubit to retrieve the uncollapsed eigenvalue pair, and whereinthe learning component utilizes the stochastic inference to determinethe expectation value further based on the uncollapsed eigenvalue pairretrieved by the second ancilla qubit.
 9. The system of claim 7, whereinthe measuring component produces an output comprising a probabilisticmeasurement based on at least one ancilla qubit. 10-25. (canceled)
 26. Acomputer-implemented method, comprising: determining, by a systemoperatively coupled to a processor and memory, utilizing stochasticinference, an expectation value during a first time interval based on anuncollapsed eigenvalue pair retrieved by an active learning process,wherein the first time interval is less than a second time intervalwhere the uncollapsed eigenvalue pair is retrieved without use of theactive learning process; and encoding, by the system, the expectationvalue associated with a quantum state.
 27. The computer-implementedmethod of claim 26, wherein the encoding comprises encoding theexpectation value as a phase.
 28. The computer-implemented method ofclaim 26, wherein the stochastic inference comprises Bayesian learningof an eigenvalue corresponding to the uncollapsed eigenvalue pair. 29.The computer-implemented method of claim 26, wherein the encodingcomprises encoding the expectation value based on an amplitude of theexpectation value.
 30. The computer-implemented method of claim 26,further comprising utilizing, by the system, a first ancilla qubit toretrieve the uncollapsed eigenvalue pair.
 31. The computer-implementedmethod of claim 30, further comprising utilizing, by the system, asecond ancilla qubit to retrieve the uncollapsed eigenvalue pair, andwherein the determining the expectation value is further based on theuncollapsed eigenvalue pair retrieved by the second ancilla qubit. 32.The computer-implemented method of claim 30, further comprisingproducing, by the system, an output comprising a probabilisticmeasurement based on at least one ancilla qubit.
 33. A computer programproduct for quantum state estimation, the computer program productcomprising a non-transitory computer readable medium having programinstructions embodied therewith, the program instructions executable bya processor to cause the processor to: determine, utilizing stochasticinference, an expectation value during a first time interval based on anuncollapsed eigenvalue pair retrieved by an active learning process,wherein the first time interval is less than a second time intervalwhere the uncollapsed eigenvalue pair is retrieved without use of theactive learning process; and encode the expectation value associatedwith a quantum state.
 34. The computer program product of claim 33,wherein the expectation value is encoded as a phase.
 34. The computerprogram product of claim 33, wherein the stochastic inference comprisesBayesian learning of an eigenvalue corresponding to the uncollapsedeigenvalue pair.
 35. The computer program product of claim 33, whereinthe expectation value is encoded based on an amplitude of theexpectation value.
 36. The computer program product of claim 33, whereinthe program instructions are further executable by the processor tocause the processor to: utilize a first ancilla qubit to retrieve theuncollapsed eigenvalue pair.
 37. The computer program product of claim33, wherein the program instructions are further executable by theprocessor to cause the processor to: utilize a second ancilla qubit toretrieve the uncollapsed eigenvalue pair, and wherein the expectationvalue is determined further based on the uncollapsed eigenvalue pairretrieved by the second ancilla qubit.
 38. The computer program productof claim 33, wherein the program instructions are further executable bythe processor to cause the processor to: produce an output comprising aprobabilistic measurement based on at least one ancilla qubit.
 39. Asystem comprising: a memory that stores computer executable components,and a processor that executes the computer executable components storedin the memory, wherein the computer executable components comprise: ameasuring component that utilizes a first ancilla qubit and a secondancilla qubit to retrieve an uncollapsed eigenvalue pair; a learningcomponent that utilizes stochastic inference to determine an expectationvalue based on the uncollapsed eigenvalue pair retrieved by the firstancilla qubit and the second ancilla qubit; and an encoding componentthat encodes the expectation value associated with a quantum state. 40.The system of claim 39, wherein the measuring component produces anoutput comprising a probabilistic measurement based on at least oneancilla qubit.
 41. A computer-implemented method, comprising: utilizing,by a system operatively coupled to a processor and memory, a firstancilla qubit and a second ancilla qubit to retrieve an uncollapsedeigenvalue pair; utilizing, by the system, stochastic inference todetermine an expectation value based on the uncollapsed eigenvalue pairretrieved by the first ancilla qubit and the second ancilla qubit; andencoding, by the system, the expectation value associated with a quantumstate.
 42. The computer-implemented method of claim 40, producing anoutput comprising a probabilistic measurement based on at least oneancilla qubit.